Non-Hermitian SSH Simulator

A computational condensed matter physics framework for studying non-Hermitian topology, PT-symmetry breaking, exceptional points, skin effects, winding topology, and complex spectra in SSH-type lattice Hamiltonians.

This project explores modern non-Hermitian quantum matter using tight-binding methods, spectral analysis, topological invariants, and scientific visualization.

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Features

• Non-Hermitian SSH Hamiltonian
• PT-symmetry breaking analysis
• Exceptional point evolution
• Complex energy spectra
• Non-Hermitian skin effect
• Edge-state localization
• Winding topology
• Animated spectral evolution
• Scientific visualization framework

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Physics Background

Non-Hermitian systems describe open quantum systems with:

• gain/loss
• dissipation
• asymmetric hopping
• nonreciprocal transport

Such systems exhibit fundamentally new phenomena absent in Hermitian quantum mechanics:

• exceptional points
• complex spectra
• non-Bloch topology
• skin localization
• PT-symmetry breaking

The non-Hermitian SSH model is one of the most important paradigmatic models in modern topological condensed matter physics.

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Repository Structure

non-hermitian-ssh-simulator/
│
├── animations/
│
├── figures/
│
├── animate.py
├── exceptional_points.py
├── hamiltonian.py
├── main.py
├── pt_phase_diagram.py
├── requirements.txt
├── run-all.py
├── skin_effect.py
├── skin_modes.py
├── spectrum.py
├── topology.py
└── winding_plot.py

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Implemented Physics

Non-Hermitian SSH Hamiltonian

The generalized non-Hermitian SSH Hamiltonian is given by:

$$ H = \sum_n \left[ (t_1 + \gamma) c_{n,A}^\dagger c_{n,B} + (t_1 - \gamma) c_{n,B}^\dagger c_{n,A} \right] + \sum_n \left[ t_2 c_{n,B}^\dagger c_{n+1,A} + t_2 c_{n+1,A}^\dagger c_{n,B} \right] $$

where:

• $t_1$ : intracell hopping
• $t_2$ : intercell hopping
• $\gamma$ : non-Hermitian gain/loss parameter

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PT Symmetry

The system exhibits PT symmetry when:

$$ [PT,H] = 0 $$

PT symmetry breaking leads to:

• complex eigenvalue formation
• exceptional points
• spectral bifurcation

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Exceptional Points

Exceptional points occur when both:

• eigenvalues coalesce
• eigenvectors coalesce

These are unique non-Hermitian degeneracies.

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Winding Topology

The topological invariant is computed using:

$$ W = \frac{1}{2\pi i} \int_{BZ} dk , \partial_k \log h(k) $$

where:

• $h(k)$ is the complex Bloch Hamiltonian

The winding number characterizes non-Hermitian topological phases.

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Non-Hermitian Skin Effect

The simulator demonstrates:

• boundary-localized bulk states
• non-Bloch localization
• asymmetric spectral accumulation

A hallmark of non-Hermitian topology.

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Installation

Clone the repository:

git clone https://github.com/akshuattri/non-hermitian-ssh-simulator.git

Install dependencies:

pip install -r requirements.txt

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Requirements

numpy
scipy
matplotlib

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Usage

Run all simulations:

python3 run-all.py

or individually:

python3 spectrum.py

python3 exceptional_points.py

python3 skin_effect.py

python3 topology.py

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Generated Results

The framework generates:

• Complex energy spectra
• PT phase diagrams
• Exceptional point evolution
• Winding topology plots
• Skin-effect localization
• Edge-state visualizations
• Spectral animations

Outputs are stored in:

figures/
animations/

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Example Results

Complex Energy Spectrum

• Real and imaginary spectral branches
• PT symmetry breaking
• Complex eigenvalue evolution

Exceptional Point Dynamics

• Eigenvalue coalescence
• Non-Hermitian criticality

Non-Hermitian Skin Effect

• Exponential edge localization
• Non-Bloch bulk behavior

Winding Topology

• Complex topological phase structure
• Momentum-space winding

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Future Extensions

Planned additions include:

• Non-Bloch band theory
• Generalized Brillouin zone methods
• Floquet non-Hermitian systems
• Higher-order topology
• Non-Hermitian Chern insulators
• Quantum transport in open systems
• Interacting non-Hermitian models

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References

1. Su, Schrieffer, Heeger,
   Solitons in Polyacetylene,
   Phys. Rev. Lett. 42, 1698 (1979)

2. C. M. Bender,
   Making Sense of Non-Hermitian Hamiltonians,
   Rep. Prog. Phys. 70, 947 (2007)

3. S. Yao and Z. Wang,
   Edge States and Topological Invariants of Non-Hermitian Systems,
   Phys. Rev. Lett. 121, 086803 (2018)

4. K. Kawabata et al.,
   Symmetry and Topology in Non-Hermitian Physics,
   Phys. Rev. X 9, 041015 (2019)

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Author

Akshu Attri
Computational Condensed Matter Physics
Indian Institute of Science